- #1

space-time

- 218

- 4

If you have a parameterized curve ξ(s) (where s is the parameter to the curve ξ) on some interval (s

_{1}, s

_{2}), then...

ξ is timelike if g

_{ab}ξ

^{'a}ξ

^{'b}< 0 for all s (just so you know, those are derivatives of the curve w/ respect to s)

ξ is spacelike if g

_{ab}ξ

^{'a}ξ

^{'b}> 0 for all s

Now, I personally wanted to see an example of a timelike curve (specifically a closed timelike curve), and I know that the Godel metric has closed timelike curves. That is why I did some calculations involving the Godel metric. Now here is what I did:

Firstly, here is the spacetime interval for the Godel metric:

ds

^{2}= (1/2ω

^{2})[ -(cdt + e

^{x}dz)

^{2}+ dx

^{2}+ dy

^{2}+ (1/2)e

^{2x}dz

^{2}]

Now here are the non-zero metric tensor components:

g

_{00}= (-1/2ω

^{2})

g

_{03}= g

_{30}= (-e

^{x}/2ω

^{2})

g

_{11}= g

_{22}= (1/2ω

^{2})

g

_{33}= (-e

^{2x}/4ω

^{2})

Every other element is 0.

Now here is the curve ξ(s) that I defined and parameterized:

ξ is the top face of a cylinder (standing upright). The top face is at height z = 10 and the radius of the cylinder is 4. The top face of the cylinder is traced counter-clockwise over the interval (0 , 2π) (which basically means that our curve is simply a circle of radius 4). Having said all of this, here is the parameterization:

ξ

^{a}(s) = [ct , 4cos(s), 4sin(s), 10] (I did not specify a t value because the t value shouldn't matter here as far as I can tell since t does not depend on s. Technically the z value shouldn't matter either since it does not depend on s).

It follows then that the derivative of this vector with respect to s is:

ξ

^{'a}(s) = [0 , -4sin(s), 4cos(s), 0]

Now then, when I do the summation of g

_{ab}ξ

^{'a}ξ

^{'b}, I get:

(16 / 2ω

^{2})[sin

^{2}(s) + cos

^{2}(s)] = 8 / ω

^{2}

That final value of 8 / ω

^{2}is positive for all s.

Now according to what I stated earlier, this would make my curve a space-like curve (not a time-like curve). Furthermore, the circular face of a cylinder is definitely a closed curve if your interval is (0 , 2π).

It would seem to me then, that I have specified a closed space-like curve. I was expecting (hoping for) a closed time-like curve, and I had read that the Godel metric apparently has CTC's running through every event.

That is why I want to ask and verify:

Have I come to the correct understanding with the work that I have done? Specifically, have I accurately specified an example of a closed space-like curve in a Godel spacetime, or did I do something wrong? Is my parameterization wrong? Is my curve just a "bad curve" or something like that? Did I simply make an arithmetic error that anyone sees?

I would appreciate any assistance. Thank you.